Cook spaghetti according to package directions in salted water until al dente. Drain and set aside.
While pasta cooks, season chicken with Cajun seasoning and smoked paprika.
In a large skillet, heat olive oil over medium-high heat. Add chicken and cook until browned and cooked through, about 6–8 minutes. Remove from skillet and set aside.
In the same skillet, melt butter over medium heat. Add minced garlic and sauté for 1 minute until fragrant.
Sprinkle in the flour and whisk continuously for 1–2 minutes to form a roux.
Gradually whisk in the milk and chicken broth, stirring constantly to avoid lumps. Bring to a gentle simmer.
Add cream cheese and 1 cup of mozzarella, stirring until melted and smooth.
Stir in Parmesan cheese and season with salt, pepper, and red pepper flakes if using.
Add cooked chicken back into the sauce and simmer for 2–3 minutes to heat through.
Add cooked spaghetti to the skillet and toss to coat in the sauce.
Sprinkle remaining 1 cup mozzarella on top, cover, and let sit for 2–3 minutes until cheese melts.
Serve hot, garnished with fresh parsley or extra Parmesan if desired.